What is the average rate of change of $f(x)=x^2+5x$ over the interval $[1,5]$ ?
Explanation: This is the formula for the average rate of change of a function $f$ over the interval $[a,b]$ : $\dfrac{f(b)-f(a)}{b-a}$ We will need to know the values of $f(1)$ and $f(5)$ to find the slope. $\begin{aligned} f(1)&=(1)^2+5(1) \\\\ &=6 \\\\\\ f(5)&=(5)^2+5(5) \\\\ &=50 \\\\\\ \dfrac{f(5)-f(1)}{5-1}&=\dfrac{50-6}{4} \\\\ &=11 \end{aligned}$ The average rate of change of $f$ over the interval $[1,5]$ is $11$. Notice that the average rate of change is calculated just like the slope of the secant line that intersects the graph of the function at the interval's endpoints. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${5}$ ${10}$ ${15}$ ${20}$ ${25}$ ${30}$ ${35}$ ${40}$ ${45}$ ${50}$ ${55}$ ${60}$ ${65}$ $y$ $x$ $(1,f(1))$ $(5,f(5))$ secant line